1. Technical Field
The present invention is directed to filtering. More particularly, the present invention is directed to a method of velocity filtering seismic signals.
2. Background Information
One technique known in the art for studying underground formations is the "Vertical Seismic Profile" technique, in which a seismic wave detector is placed in a borehole successively at different depths, with seismic waves being emitted from a source on the surface and with the signals produced by the detector being recorded by the seismic wave detector located downhole.
The essential purpose of these measurements is to determine the reflective horizons or reflectors which are situated deeper than the bottom of the borehole by analyzing the waves reflected on these reflectors and rising towards the detector. These waves are commonly referred to in the art as "upgoing" waves.
The detected waves comprise not only the upgoing waves, but also waves known as "downgoing" waves, which include waves that have propagated directly from the source to the detector, waves which have been subjected to multiple reflections, and interfering waves of various kinds.
By putting together all of the recorded signals in a single document, it is possible to detect coherencies between the various traces. However, given the multiplicity of superposed components in each signal, such a document is extremely difficult to interpret.
In order to detect reflectors and their positions, it is therefore desirable to filter the recorded signals in order to analyze the upgoing waves.
One prior art filter method for reinforcing a given wave component in a set of signals recorded at levels z.sub.i (where i lies between 1 and n) is described in French patent document FR-A-2 494 450, herein incorporated by reference. This filtering consists in advancing the m (m&lt;n) first signals g.sub.i (t) recorded at levels z.sub.i by a time t.sub.i where 2&lt;i&lt;m relative to the signal g.sub.1 (t) recorded at level z.sub.1 in order to align the downgoing wave components and produce a first signal by adding together the signals shifted in this way. Similarly, a second signal is produced by adding together recorded signals g.sub.i (t) for 1&lt;i&lt;m after delaying the signals g.sub.i (t) for 2&lt;i&lt;m by a time t.sub.i relative to the signal g.sub.1 (t). The first and second signals are then combined in order to produce a signal u.sub.1 * representative of an optimum estimation of the upgoing wave component. Signals u.sub.2 *, . . . , u.sub.k * are produced in the same manner from set of m recorded signals {g.sub.2 (t), . . . , g.sub.m+1 (t)}, . . . , {g.sub.k (t), . . . , g.sub.m+k-1 (t)}, . . . .
This method suffers from the drawback of reinforcing, in practice, only those waves which are aligned in the recorded signals after they have been shifted, whereas waves having different velocities simply do not show up.
Another prior art filter technique is the F-K velocity filter method which serves to separate upgoing waves from downgoing waves. This method is described in the article entitled "Vertical Seismic Profiling" by B. A. Hardage, Geophysical Press, 1983, pp. 175-179, herein incorporated by reference.
In the F-K velocity filter method, the set of signals g.sub.i (t) recorded by the detector at depths z.sub.i for 1&lt;i&lt;n is taken to be a two dimensional signal g(z,t) and the following successive operations are performed:
(a) the two-dimensional Fourier transform G(k,w) of the signal g(z,t) is calculated where k and w are the dual variables of z and t respectively, in the Fourier transform. The signal G(k,w) has the advantage that the upgoing waves are to be found in the quadrant k&gt;0, w&gt;0 and the downgoing waves are in the quadrant k&lt;0, w&gt;0;
(b) the coefficients contained in the quadrant k&lt;0, w&gt;0 are set to zero or are multiplied by a small factor, e.g. 10.sup.-3 ; and
(c) the inverse two-dimensional Fourier transform is calculated and in the resulting signal g.sub.u (z,t) the downgoing waves are highly attenuated.
This method suffers from problems due to side-effects because of the sharp cut-off between those coefficients which are modified in operation (b) and the other coefficients which are not modified. In particular, this gives rise to artifacts and to spectrum folding during operation (c).
Further, the Fourier transform using each of the variables z and t requires a certain number of samples to be available. This is no problem for the variable t since there are generally several thousand points available along the t-axis, given that each signal is recorded over a period of several seconds and that the sampling period is about 1 ms. In contrast, the number of points available along the z-axis depends on the number of levels at which signals are recorded, and this is typically much less.
In order to perform the Fourier transform of the variable z, it is necessary to have at least 64 different recording levels, with the gap between consecutive levels being no greater than a value related to the highest frequency in the received signal and to the lowest propagation velocity, with a typical value being about 10 meters (m). Such a procedure is too expensive.
Another prior art velocity filter is the Tau-P method. This method is equivalent to the F-K method but has the advantage of being suitable for implementation with a smaller number of signals, thereby reducing measurement costs. However, this method suffers from the same defects of artifacts and spectrum folding as the F-K method.